Modern Methods in Computational Molecular Thermodynamics and Kinetics
Modern Methods in Computational Molecular Thermodynamics and Kinetics CBE 60547
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Lecture 9 Plane waves and core potentials 1 Hydrogen atom in a box Recall H atom problem Vr 1r e ex ctanswer ofbasis funcEOnS 39 asuttueie emu Xx 5 m123 a a n n l 57 39 21nd Pamela m a bux suluttuns tut an sleetmn m at nm bux Could express H atom solutions as linear combinations ofthese Not particularly ef cient but if mm Keep only even n anorewnte waves as 1 1 2m Xx 2 G mHt2 a r 035 mm ecampmamicnmsuy 112 Unmstyamam mm me w F Schmldex ll m m PM llmuug Lecture 9 Plane waves and core potentials Ditto y and 2 Can write solutions for H atom as linear combination of these plane wave basis functions 27rm 1 i x Zr x zcimecn s Gm 12202 thZ Can t keep in nite number of m so have to cut off somewhere de ned by Tquot lt 2 Emuff m m De nes minimum wavelength retained in basis and thus size m of basis Note for a given Ecumff larger at implies bigger m Within DFT if we de ne a box size an Ecumff and a particular exchangecorrelation functional we ve got ourselves a model 1 a2 5 1x 05mm x 0 70631 W x 6 W x x Kinetic energy Diagonal in basis functionseasy 2 1 d eianxgt Gm 1 6M em 24172 Potential energy terms Can take advantage of Fourier transforms to evaluate 1x 20Gme39Gquot vGM lJvxequotquotquotdx evaluate on a realspace FFT grid a era lt6an 0x eiGxgt 2 vGM lt6an e G quot gt we 2 Periodic boundary conditions Subtle but important point is that this approach is based on a periodic representation of a system an arti cial construction for something like an atom or molecule that isn t actually periodic Use the term supercell to describe periodic box Have to make box large enough to avoid spurious interactions between periodic images In particular 1x veaulamb x is periodic and longranged because it contains all the electrostatic lr terms Have to use special tricks Ewald summations to evaluate these sums and have to group electrostatic terms to avoid nonconvergent sums Key limitations of supercell approach see Martin 0 Supercell must be net charge neutral the electrostatic energy of an in nite charged system diverges Supercell must not have a net electric field The absolute electrostatic potential is not wellde ned there is no vacuum to reference an electron energy to in an in nite system Prof W F Schneider CBE 60547 7 Computational Chemistry 212 110707 PM 1122009 University ofNotre Dame Full 7 Lecture 9 Plane waves and core potentials It is possible to overcome some of these limitations by introducing compensating background charges dipoles multipoles 3 Supercells Cartesian and fractional coordinates Planewave calculations use periodic boundary conditions Have to define two things to describe atomic arrangement 1 Lattice constants of periodic cell defined by three lattice vectors a1 a2 213 e g our cube for H but could be any of the Bravais lattices depending on the relationships between the three vectors Every point R is equivalent to any other point R39 R mal nzaz 71333 2 Locations of atoms within the periodic cell the socalled basis Latter can be done by specifying locations of atoms in Cartesian coordinates 1 Typically easier to specify in fractional coordinates f of the lattice vectors Related by 1 Af where A is matrix generated by combining lattice vectors in column form 4 Gaussian v Vasp Compare Gaussian 63 1 lGdp and Vasp inputs and outputs for a spinpolarized PW9l H atom Gaussian Vasp 1 input file 4 input files POSCAR structure input INCAR program options POTCAR identities of atoms KPOINTS kpoint sampling more later zmatriX coordinates Atom positions relative to a supercell Arbitrary basis set Planewave cutoff Detailed options hidden Options out there Hartree units eV units 2 output files log and chk w of output files OSZICAR iteration summary OUTCAR detailed output CONTCAR final geometry Prof W F Schneider CBE 60547 7 Computational Chemistry 312 110707 PM 1122009 University ofNotre Dame Full 7 Lecture 9 Plane waves and core potentials 6 basis functions 6311Gdp 9045 basis functions 10 A cube on a side 250 eV cutoff 3 cycles to converge good initial 15 cycles to converge initial plane wave guess more guess difficult Spinup and spindown orbital Spincomponent band energies eigenvalues Total energy referenced to ionized Total energy referenced to pseudopotential atomic state atom General things going on in Vasp calculation very similar to Gaussian but details differ considerably set up Hamiltonian H p in o wn and pin re nement of density DHS algorithm 39 a P Pulay Chem Phys Lett 73 i new charge denSIty p01 in lwnh l 393 1980 iterative re nements ofwave lnctimisWu re nement of density pm pm gt new pm o re nement of wavefunctions blocked Davidson like algorithm no W calculate forces update ions ke J I KELEESE PEELOPOTEI NP 1 Page IS Vasp uses convention in POSCAR that atoms be speci ed in groups of like type and that the order in the POSCAR corresponds to the order of atoms in a composite POTCAR 5 Core electron treatments What is hiding in the POTCAR A description of the atomic frozen cores Plane waves are poor choice for treating wavefunctions near atomic cores Rapid oscillations 9 high kinetic energy 9 high m plane waves Prof W F Schneider CBE 60547 7 Computational Chemistry 110707 PM 1122009 University of Notre Dame Full 7 n Lecture 9 Plane waves and core potentials one approach is to augmm the plane waves APW with functions that better represent the atomic characte of orbitals in spheical regions around the atoms In geneal quite expensive approach Moden linearized augnnented plane wave LAPW approach makes it tractable still not in broad use Alternative is to replace explicit treatment of cores with implicit treatment that captures the effects ofthe atomic cores on the valence orbitals Pseudopotential models Idea is to replace core electrons and potential with pseudapm m39al Establish radial cutoff between core and valence regions choose specific atomic configurations construct PP to go smoothly into real potential outside of cutoffradius preseves phase and shape of valence wavefunctions amide cutoff cmquot radius Sumdaldlwggn Transfeability e ability of PP to be applied in diffeent chenical environments Local vs nonlocal e Spherically symmetric local vs 1depeident nonlocal conseving e Preseving M1 of valence wavefunctions important for getting longrange Coulomb potential correct Soft vs hard 7 Hard close to true atomic potential requiring large KE cutoff Soft large peturbation on atomic potential allowing smalle KE cutoff Ultrasoft c 39 39 quot 39 39 39 39 39 and compensate by putting augmentation charges on all atoms Pro39ecmr au aimed wave A PAW most modern approach combines advantages ofAPW and ofultrasoft Pps Not strictly a PP approach construce full wavefunction as combination ofvalence plane wave part an A 39 39 39 c potential 39 quot 39 39 structure ofvalence wavefunctions Generation of these potentials is not for the casual use just as the case for making AO basis sets The eat strength of Vmp is a vey complete and reliable library of potentials for the whole peiodic table and for seveal exchangecorrelation functional 21amng Relativistic effects are greatest in the core regions massvelocity Darwin spinorbit All these approaches allow the relativistic effects to be treated for the core states so that the indirect ects prop agate into the valence me w F Schnada39 CEE EISMrCumputauunal chemistry 512 ll n7 n7 PM llZZEIEID UnvesityquutreDame Fall nnm Lecture 9 Plane waves and core potentials Following slides borrowed from Georg Kresse presentation Pseudopoten al approximation ultrasoft pseudopotentials PAW potentials G Kersss PEEL DOFOTEK HALS PART i1 normconserving pseudopotentials the course of the remaining calculations a three different types of potentials are supported by VASP they will be discussed in more details in later sessions a the number of plane waves would exceed any practicle limits except for H and Li all three methods have in common that they are presently frozen core methods ie the core electrons are precalculated in an atomic environment and kept frozen in J Page 11 Prof W F Schneider 110707 PM 1122009 0 9 5 15 C 3 g 10 E o a 4 an exact potential interstitial region pseudopotential Al effectiv A1 atom PAW A1 atom 3P 2P 3P 3 s 1 s 3 s 2p and Is are nodal structure nodeless I is retained S G mm PsiL noPorENnAu Pu I Umverslty of Notre Dame Fm 700 tannin cm a W p ru an an CBE 60547 7 Computational Chemistry Page 12 612 Lecture 9 Plane waves and core potentials character of waveflmctiou HIquot PAW addidan39ve augmentation 2 l l hns lme 1quot 2 mime Chm PP I quot pseudo pseudowusile AEousite 0 same trick works for exchange correlation energy Hartree energy Page wave mctious G RIBS PSEUDDFo r HALS What U AND PAW to deal with long range electrostatic interactions between spheres is introd similar to FLAPW charge density kinetic energy Hartree energy a the pseudowavefunctions do not have the same noun as the AE wavefnnctions inside the spheres o C pseudo compens pseudocomp onsite AEonslte I Hartree energy becomes EH E 7E1 E1 l compensation charge at site mus Fur n in ma 1 pseudocharge at one site amass Psmmmwo CBE 60547 7C0mputati0nal Chemistry iver 39 Notre Dame 397 Prof W F Schneider 1 10707 PM 1 122009 Pig 75 712 Lecture 9 Plane waves and core potentials W PAW enelgyfzmcrional RE Brecht Phys Rev B50 7953 0994 0 total energy becomes a sum of tlu39ee terms E E E l 1 1 t e E 2mm 7 3At 71gtEwtnnrvq 7 mm fvgt zc mm rlltrn d31 mute l A e r E 2 2 pU t 5AM Ecn nnc sites 1th 5me 19 VHlnzcj71t r 131 9 1 a E 2 2 Pijlt rl 5AM Ex cl 11 11 sites ij EHlle VHlllzcllll d3139 1 a mass Psznoemxnus mm mm PAW Page as n E is evaluated on a regular grid with additional compensation charges to account for the incorrect norm of the pseudowave mction very similar to ultraso pseudopotentials 7 anh iln il pseudo charge density 1 compensation charge a E1 and E 1 are evaluated on radial grids centered around each ion these terms correct for the shape difference between the pseudo and AE wavefunctions o no crossterms between plane wave part and radial grids exist k j 5 mass Primayum mus Pu 11 Arm mm Page 21 o Prof w F Schneider CBE 50547 7 Computational Chemistry 812 U 110707 PM 1122009 ersity ofNotre Dame mu mm Lecture 9 F N US PP or PAW potentials the general rule is to use PAW potentials wherever possible I less parameters involved in the construction of PAW potentials magnetic materials a alkali and alkali earth elements early 3d elements to left of periodic table W certainly easier than for USPP general construction scheme is similar for USPP and PAW potentials I nrost of the PAW potentials were generated 5 years after the USPP a different philosophy c mess r5 PAW Am User Bangui page 3 Pseudopotentials 7 NC U SPP PAW PAW versus US PP potentials the PAW potentials are generally of similar hardness across the periodic table for the USPP the radial cutoff were chose according to the covalent radius a most of the PAW potentials were opti Permdic Table nrised to work at a cutolif of 250300 a the USPP become progressively eV softer when you move down in the o PAW potentials are usually slightly P iOdlClable harder than USPP c for compounds where often species with very different covalent radii are mixed the PAW potentials are clearly superior c for one component systems the US PP might be slightly faster at the price of a somewhat reduced precision G K1555 m MWAL39D our was was Prof W F Schneider 110707 PM 1122009 CBE 50547 7 Computational Chemistry University ofNotIe Dame mu 7 n Plane waves and core potentials 912 Lecture 9 Plane waves and core potentials Pade approximation I Irecmninend to use either the 1 description is not that strict 9 Russ 25 Pm n L39SVPP mums download location of PW91 potentials paw GApotcai three different avours one LDA CA and two GGA s PW91 and PBE 0 download location of LDA potentials pawpotcar date tar clate tar o download location ofPBE potentials panEEpotcar date tar the PBE implementation follows strictly the PBE prescription whereas the PW91 for the LDA part the parametiisation of Perdew and Zunger is used instead of Perdews for the PBE potentials you do not need to specify VOSKOWN1 in the INCAR le since this is the default Page g 1012 Standard PAW potentials and Energy Cuto iv BJJ 700 CJJ 700 Nil 700 0J1 700 FJl 700 B 318 C 400 N 400 O 400 F 400 EA 250 C 273 Ns 250 05 250 F5 250 A1 240 Si 245 P 270 S 280 C1 280 Al11 295 Sid 380 PJJ 390 5J1 402 ClJl 409 Ga 134 Ge 173 As 208 Se 211 13139 216 G d 282 Geil 287 GaJl 404 Gel 410 In 95 Sn 103 Sb 172 Te 174 1 175 IlLd 739 SJLd 241 T1 90 Pb 98 Bi 105 T141 237 Pbd 23 7 BLd 242 K c mg n Prof W F Schneider CBE 60547 7 Computational Chemistry 110707 PM 1122009 Uni 39 fNotIeDame mu mm Lecture 9 Plane waves and core potentials f PAW hard AE H 1447 1446 Li 5120 5120 Bel 4520 4521 results for the bond length of several Naz 5663 567 quot molecules obtained with the PAW and AE CO 2141 2128 2129 approaChes N2 2076 2068 2068 I using standard PAW potentials and hard F2 2633 2621 2615 PAW potentials 35721 o well converged relaxed core AE calcula L835 1833 tions yield identical results 1048 1050 1 2470 2464b 7948 2945 Wotan Phys 99 3898 1993 1 GAUSSIAN94 S Goedecker et al Phys Rev B 54 1703 1996 m um m 16 The energy zero in VASP all energies are referred to the the reference state for which the potential was generated this is in most cases not the real groundstate of the atom I to determined the energy of the grounstate of the atom place the atom in a larger non cubic box to break initial symmetry Le 11 AX lO AX 9 A use the F point only INCAR ISPIN 2 1 spin polarized ISMEAR 0 SIGMA 02 1 for small sigma conv for TM is diff MAGMOM 2 1 initial magnetic moment one should use the energy value energy without entropy of the OUTCAR le since this coverges most rapidly to the correct energy for sigmagt0 39 Ecol EmetaImolecule quotEatnm J 395er am ASE Page 26 6 Planewave cutoffs Core potentials typically generated to be used with some minimum cutoff as indicated above Advantage o planewave calculations is that basis can be improved systematically by simply increasing cutof o Prof w F Schneider CBE 50547 7 Computational Chemistry 1 l12 110707 PM 1122009 U iversity ofNotIe Dame a mm Lecture 9 Plane waves and core potentials When comparing energies between different calculations eg to calculate reaction energies must make sure to use the same cutoff for all calculations 7 Optimizations Forces calculated using HellmanFeynman theorem No concern about Pulay forces Optimizations generally limited to Cartesian coordinates in PW calculations Show example of N2 optimization What is N N distance Prof W F Schneider CBE 60547 7 Computational Chemistry 1212 110707 PM 1122009 University ofNotre Dame Full 700 Lecture 4 Practical Aspects of Electronic Structure 2 1 BornOppenheimer Approximation Need to elaborate our models to handle molecules More particles more variables to keep track of For Hz for instance would have to solve for energy E and wavefunction Yrlr2RnRb including all four particles Here r refer to the coordinates of the two electrons and R to those of the two nuclei All have kinetic energy and interact via Coulomb forces Further in principle we d have to make sure the wavefunction obeys the Pauli principle for electrons and all identical nuclei Molecular Schrodinger equation Nuclear Electron Electron electron electron Nucl earnucl ear KE Nuclear KE attraction repulsion repu sion 2 2N nsznnzNNZZez L2vgi iv QZLHE 2 12 2 A Yr1rR1RN 2mg 1 2 0 Ma 1 a rig x 1xl ij a an Ra EYrlrnRlRN Quite a mess Masses of nucleiM are always gtgt m2 Electrons will move much faster than nuclei and nuclei instantaneously will appear immobile The electron and nuclear motions can thus be approximately decoupled by separation of variables Essence of Born Oppenheimer approximation YrrRRN Ilwo39lr R1RleR1RN Substituting we get electronic Schrodinger equation which depends only parametrically on the locations of the nuclei n A n n 2 2 ipmrprRRN Emlyn 13rRRN 1ry39 quotZez Same ideas about orbital approximation I rlrn ltplr11pn rngt and applying variational principle to give effective UCoulombpUexchangevconelaionl1pj1pi fill Siwi In all approximations veaulambpfprzdrz pr 2111120 12 I In HartreeFock vexchange is treated exactly and vconela on ignored In HartreeFockSlater vexchange is approximated and vconemion ignored In more advanced methods Prof W F Schneider CBE 60547 7 Computational Chemistry ll 9152009 122323 PM University of Notre Dame Fall 200 Lecture 4 Practical Aspects of Electronic Structure Each of these points in 3N In all cases we ultimately end up with dimensional space is 3 equations that look exactly like our equations soluti on to electronic for atoms except that the oneelectron part SChrod39quotser aqua hi now includes Coulomb attractions to all the A c nuclei instead of Just one De nes an adiabatic potential energy surface PES N N z z e2 R EmRlRNE 2 2 7quot a a1 Ra EFES B Nuclei can be thought to travel on this PES 5 Can be treated several ways A c A c 1 Nuclear motion could be described quantum mechanically eg to capture tunneling Expensive and specialized 2 Can be described as classical particles rolling along the PES Essence of classical ab initio molecular dynamics 3 Can focus on locating critical points along PES like stable minima molecules and saddle points transition states Least expensive and most common in computational chemistry BomOppenheimer approximation is pretty robust by and large 2 Bring back the basis sets The oneelectron HF or HFS equations give us de ning equations for the energyoptimal orbitals but they aren t convenient to solve for anything more complicated than an atom What to do Reintroduce idea ofa basis set IMO E Cyi p r This is perfect if the by form a complete basis which of course is never possible Trick again will be to nd a convenient and sufficiently complete one If basis functions are atomiclike often called LCAO MO approximation or linear combination of atomic orbitalsmolecular orbital approximation Substitute winto one electron equations multiplying and integrating gives EFWCV Q2 swcv or FC SCs V V where F and S de ne the Hamiltonian Fock and overlap matrices just like we had when we developed the secular equations F lt i vgt S 45 Translates problem from integrodifferential equation into a secular matrix problem Dif culty is that the Fock matrix elements depends on all the 11 2 Cum u In HartreeFock case called HartreeFockRoothaan equation Substituting again Prof W F Schneider CBE 60547 7 Computational Chemistry 22 9152009 122323 PM University of Notre Dame Fall 200 Lecture 4 Practical Aspects of Electronic Structure gt lt at Hm 2 2pm uvLa Mav Differs only in details for HFS or DFT This simpli es things down to a charge density matrix Ki ii Jj F lt gtN22 N12 PM 22 CMij which expresses the charge density in terms of the basis pr 2 2m an r and a bunch of two electron integrals 1 W llafj 1 vlt1g ilt2 a 2dr1dr2 The two electron integrals are determined by the basis The density matrix has to be guessed and then updated with new coef cients during SCF cycles It is this matrix that is usually converged upon Here s our new self consistent field algorithm 1 Select a basis set 2 Calculate oneelectron and twoelectron integrals only once 3 Guess some coef cients Cdensity matrix Pdensity p This guess matters It determines what electronic con guration you are calculating Construct and solve Fock matrix for coef cients C and eigenvalues 3 Construct new density matrix P39 and compare to old Difference greater than tolerance update P and return to Step 4 Difference less than tolerance all done Check to see whether you arrived at the electronic con guration you wanted 399 3 Semiempirical methods A HartreeFock calculation roughly scales in computational expense as M where N is the number of electrons Depending on your computer and the size of the system you want to model this can be prohibitively expensive Many methods developed to simplify the calculations See Cramer for a nice description Extended Huckel Theory EHT Uses minimal basis of Slater orbitals uses empirical rules to construct Hamiltonian matrix elements by tting to some experimental constants Noniterative useful for molecular orbital analysis won Robert Woodward and Roald Hoffmann a Nobel Prize Called tight bonding in the physics world Complete Neglect ofD erential Overlap CNDO and Intermediate Neglect ofDifferential Overlap INDO Like EHT but parameterizes one and twoelectron integrals rather than matrix elements Sets S m 6M in Fock matrix Archaic Neglect of Diatomic Differential Overlap NDDO Includes more integrals Many avors including MNDO MINDO AMl PM3 All involve different parameterizations often t to experimental data Still actively developed but superseded in many applications by DFT Utility depends on particular problem Prof W F Schneider CBE 60547 7 Computational Chemistry 33 9152009 122323 PM University of Notre Dame Fall 200 Lecture 4 Practical Aspects of Electronic Structure 4 Openshell systems All this development has assumed a system with all electrons paired Much of interesting chemistry like transition metal systems or breaking chemical bonds involve unpaired electrons Theory has to elaborated to treat these Unrestricted H artree F ock UHF creates separate spinorbital for each electron All electrons interact through Coulomb interaction but for like electrons diminished through exchange interaction UHF advantage is that it is easy and convenient disadvantages are that you have twice as many orbitals to keep track of and the resultant wavefunction is not an eigenfunction of the spin operator so they are not pure spin states Described as spincontaminated Restricted Open Shell Hartree Fock ROHF retains the same spatial orbitals for spinup and spindown electrons but only singly occupies some orbitals Advantage is that solutions are pure spin functions Disadvantage is that some spin states can only be described by linear combinations of Slater determinants recall openshell examples above calculations become more cumbersome and poorer solutions than UHF for given amount of computational effort quot325 Lithium 1522sl quotR H Fquot R OH F arti 39 cial Pure Doublet singlet doublet higherorder spin contributi ans Note that minority spin UHF orbitals are higher in energy because their Coulomb interactions are diminished by fewer exchange interactions 5 Gaussian basis sets Generally seek basis functions for which the necessary integrals can be calculated conveniently Recall Slater functions eff have correct functional form near nucleus but manycenter integrals cannot be calculated analytically Gaussian functions 6quot2 turn out to have functional form particularly convenient for calculating necessary integrals Most common procedure for molecules is to write molecular orbitals in expansions in terms of Gaussian atomiclike functions centered on the atoms Many such Gaussian basis sets have been created for various purposes and considerable jargon associated with concept One of John Pople s 1998 Nobel prize in Chemistry wWalter Kohn major contributions Primitive Gaussian is one Gaussian basis function de ned by an exponent C and an angular part given by some product of x y and 2 Prof W F Schneider CBE 60547 7 Computational Chemistry 44 9152009 122323 PM University of Notre Dame Fall 200 Lecture 4 Practical Aspects of Electronic Structure 3 4 i 39k gag L15 2i w xiyjzke zr n 2i2j2k For instance i j k 0 would de ne a single spherical stype primitive Gaussian The set i39k 100 010 and 001 would de ne a set of three ptype primitive Gaussians Exponent C determines how fast or slow the basis function decays away from the atom Big C fast decay function close to nucleus small C slow decay function far from nucleus Because individual Gaussians are a pretty poor representation of the behavior of real atomic wavefunctions especially near nuclei common to bundle together or contract several Gaussians with different exponents into one basis function W 2mg m is typically a small number maybe up to six A basis set is a collection of exponents and contraction coef cients de ning a basis for an atom or atoms Constructed in many ways be tting to numerical results for atoms and for some set of properties of molecules Common notation primitive Gaussians 9 contracted functions eg 10s 5p 2d 1f 9 4s 3p 2d 1f Note a d can be either 5 or 6 functions depending on whether the symmetric component x2y2zz is included cartesian or removed spherical Similarly f can be 7 or 10 Minimal basis sets To describe an atom within a molecule would want at a minimum one basis function for each occupied atomic orbital Called a minimal basis set For instance Hydrogen 1s one basis function Fluorine ls 2s 2pX Zpy 2pZ ve basis functions Iron 1s 2s 2pxyyyz 3s 3pxyyyz 3dxy 3dxz 3dyz 3dxz 3dy2 3dzz 4s 4pmZ 19 basis functions For example STO3G is shorthand for a minimal basis set in which each basis function is a contraction of three Gaussians For uorine 6s 3p 9 2s 1p STOnG basis sets available for many atoms Good for rough and ready calculations but not very accurate Example minimal basis set HFSTO3G calculation on HF Six basis functions input six molecular orbitals output Walk through Gaussian output PopFull GFInput to see basis information Prof W F Schneider CBE 60547 7 Computational Chemistry 55 9152009 122323 PM University of Notre Dame Fall 200 Lecture 4 Practical Aspects of Electronic Structure A 16 H1s szz F2S IILUMO 126 eV 10 szx 10 szy quotHOMOquot 156 eV 070 szz 053 H15 041 FZS HFSTO3G HF 09556 A E 9857284 Hartree u 125 Debye 397 eV 095 F25 025 Fls 015 Hls EIECtrOSta L I c notentl 3 6 7049 eV 099 F15 H F gt z Note Where all the energy is in the core Note Where all the chemistry is in the valence Basis sets have to balance treatment of both good core lowers energy but may not help With chemical part Multizeta basis sets Minimal basis set doesn t leave any real room for atomic orbitals to breath expand or contract When forming bonds With other atoms Double zeta basis set one With two basis functions per occupied atomic orbital More common is split valence Which is double zeta for valence levels single zeta for core Pople basis sets very common split valence type For example 63 1G basis set for uorine ls orbital described by 6 primitive Gaussians contracted to one basis function One set of 2s and 2p orbitals described by contraction of 3 primitive Gaussians One set of 2s and 2p orbitals described by l primitive Gaussian Triple zeta and triple split valence similar for instance 631 1G Polarization functions To capture polarization of atoms that occurs When forming chemical bonds it is necessary to include basis functions of higher angular momentum than the occupied atomic orbitals e g p functions for H or a functions for C Prof W F Schneider CBE 60547 Computational Chemistry 6 9152009 122323 PM University of Notre Dame Fall 2007 Lecture 4 Practical Aspects of Electronic Structure For example 631Gdp aka 631G augments 631G with a single set of polarization functions on each atom Multiple polarization functions can be added and are particularly important to capture electron correlation effects Dunning s correlationconsistent basis sets another family that includes polarization functions in a systematic way eg ccpVDZ polarized valence double zeta is similar in size to 631Gdp ccpVTZ similar to 6311G2df2pd higher levels available Di use functions Lastly for anionic systems or loosely bound electrons common to include diffuse functions same angular momentum as occupied states but small exponents Denoted 631Gdp in Pople notation augccpVDZ in Dunning Some sample results for HF molecule from CCCBDE Hartree Fock 39 39 STO3G 6316 6316M ccpVDZ augccpVDZ 63116M ccpVTZ ii basis functions 51 92 155 155 269 206 3815 Energy Hartree 985728 999834 1000117 1000197 1000338 1000469 1000585 E HOMO eV 126 172 171 172 178 176 Dipole moment D 1252 2301 1944 1918 1899 1980 1905 Total energy drops with increasing basis size consequence of variational principle HartreeFock limit is energy in limit of complete basis Other quantities converge more quickly with basis size Some families of basis sets are constructed to facilitate extrapolation to the complete basis limit Electron cores Low energy core electrons typically don t participate much in chemical bonding but can add a lot to computational cost of HartreeFock and beyond calculations Generally seek approximations Especially important for heavy elementsmetals Basic assumption is socalled rozen core approximation that the core electrons can be assumed to be constant in different chemical environments Not strictly right but for chemical bonding typically a good approximation First need to make somewhat arbitrary decision about what is core and what is valence Within HartreeFock for instance we could separate the Fock operator like this hl E 23921 KAJ 2 2 16 511401 A 2 1 A 1 J wrzi er K n Mu102er 1110 Goal is to replace core operators on 14 with some simplertocalculate operators derived from accurate atomic calculations Simplest way not nearly good enough though would be to say that valence electrons see nucleus classically screened by core electrons Instead of valence seeing lr see diminished lr Valence wavefunctions won t have the right nodal behavior near the core but the valence part will can be ok Prof W F Schneider CBE 60547 7 Computational Chemistry 77 9152009 122323 PM University of Notre Dame Fall 200 Lecture 4 Practical Aspects of Electronic Structure 1 1 Effective Core Potentials L l1 1e 2ml 70 Years of Development 0 CS Dirk Andrae Na quot39 Theoretische Chemie lrquot Fakultat f39Lir Chemie Universit t Bielefeld 1I3939 Hellmann Memorial Meeting 1 x Bonn 2003 07 26 A l l l 0 O 1 2 3 4 5 Effective core potential 1quot739 in atomic units for sodium Na 1826 i 0536 and resiuni 7 l 1672 b 031111 used in this form by Hellmann and Kassatochkin in a study of metallic binding in alkali metals H Hellmann W Kassatotschkin Die metallische Bindung nach Clem kombinierten Naherungsverfahren Acta Physicochim U R S S 5 1936 23 44 parameters adjusted to experimental term energies To do right have to take care of exchange and the different potentials seen by electrons with different angular momenta Provides convenient opportunity also to take care of relativistic e ects which are most important near the core Massvelocity correction electrons near core move at speeds close to c increase in mass and thus decrease in energy Spinorbit coupling causes splitting into j l i s states Darwin correction could show Fig 29 from dissertation Nonrelativistic and relativistic e ective core potentials ECPs available for many elements Typically these have to be combined with basis functions designed to work with them Most common are HayWadt LANL and StevensBaschKrause SBK Other more modern ones also available See Reviews in Computational Chemistry 1996 6 Choosing basis setsECPs Of course would like a complete basis set but computational cost almost always limits basis set size Further there is no one unique way to construct atomcentered basis functions or core potentials so there is always some arbitrariness to these basis sets Much research has gone into constructing basis sets Rules of thumb 1 More basis functions are generally better than fewer Trick is to add basis exibility in regions you care about Also generally need to have balance in exibility amongst atoms 2 Consider nature of problem at hand Structures dipole moments vibrational frequencies can generally be captured with modest double zeta polarization basis Accurate absolute bond or excited state energies may require larger bases but relative energies could be achieved more modestly Anions require variational exibility far from the atom so diffuse functions N1R requires good cover near the core region Prof W F Schneider CBE 60547 7 Computational Chemistry 8 9152009 122323 PM University of Notre Dame Fall 2007 Lecture 4 Practical Aspects of Electronic Structure W Personal experience literature and convention Experience is a wonderful guide and the literature is a great resource to take advantage of other s experience Many tabulations like the CCCBDB to help guide And certainly approaches like HF631G B3LYP6311G are venerable and well calibrated In general though careful use means the user tests the sensitivity of computed results to basis set and other approximations 5 Balance of approximations Little sense to converge your results with respect to basis set beyond the reliability of the model eg HartreeFock that you are using U Consider types of atoms involved Not all standard basis sets cover all atoms 6 Computational resources ultimately dictate what you are able to do In very accurate calculations often apply extrapoloations to complete basis set CBS limit Basis set resources 0 httpwww ems nnl 39mnixfm m html Davidson and Feller Basis Set Selection for Molecular Calculations Chem Rev 1986 86 681 r Gaussian manual Performance details of SCF methods Basis is often orthonormalized to eliminate overlap from HFR equation allows equations to be solved by matrix diagonalization Initial guesses for P are obtained by solving an approximate Hamiltonian like extended Huckel Always beware Initial guess can in uence nal converged state Because the number of 2electron integrals grows as N4 they are sometimes calculated as needed onthe y called direct SCF The SCF procedure is an optimization problem As discussed above success depends on a reasonable initial guess for P and judicious updating Various strategies can be used to speed and stabilize convergence like damped mixing of previous cycles Second order SCF is a convergence acceleration method that requires calculation or estimation of the rst and secondderivatives of the energy with respect to the orbital coef cients See eg Chaban et al Theor Chem Accts 1997 97 8895 Pulay s direct inversion in the iterative subspace or DHS is a popular and powerful acceleration procedure that extrapolates from several previous Fock matrices to predict optimal next Fock to diagonalize 8 Symmetry In molecules that possess symmetry Fock matrix is simpli ed because integrals of basis functions from different symmetry classes equal zero Look at HF example above F 2pX and 2py orbitals have 7 symmetry and cannot interact directly with s basis functions on H Indirectly affected by changing charge density Other orbitals have 6 symmetry Fock matrix becomes block diagonal in these two symmetry classes Prof W F Schneider CBE 60547 7 Computational Chemistry 99 9152009 122323 PM University of Notre Dame Fall 200 Lecture 4 Practical Aspects of Electronic Structure Nonlinear molecules can construct symmetry adapted basis functions Water example 9 Population analysis The molecular orbital solutions of the HartreeFockmolecular orbital model contain lots of useful information that can be helpful in understanding structure and bonding Perhaps most direct is the distribution of charge around the molecule pr 2 w2r22Pi rgt r occupied u v Can also calculate moments of the charge dipole quadrupole Chemically it is usually more interesting to assign charge to individual atoms in what is called a population analysis There is no single right way to do this the charge or an atom in a molecule in not uniquely de ned Consider an occupied molecular orbital made up of two basis functions on two different atoms a and 8 1P ca a cp p 2 2 10 lip ca c 2cmc i Q 45 c can be interpreted as the fraction of charge in w that can be assigned to a and likewise c for 8 The remainder is the overlap population between a and 8 Who gets these In Mulliken population analysis they are divided equally between the two atoms Summing over all the occupied orbitals and subtracting from the nuclear charge gives the gross atomic populations Mulliken populations are directionally very useful but their absolute magnitudes have no real meaning They are sensitive to the choice of basis and equal distribution of overlap density between atoms is rather arbitrary The overlap problem goes away if the basis is orthogonal so the overlap terms go to zero In the Lowdin approach the basis set is orthogonalized according to by 2839 V so that lt4 quot gt 6m MO V VI coef cients in this transformed basis used to assign charge to individual atoms More stable less common than Mulliken approach This orthogonalization scheme is not unique so other implementations possible Natural orbitals are another more sophisticated scheme for orthogonalizing and assigning charge See httpwwwchemwiscedunboS Again more stable than Mulliken more information rich Bader analysis another more sophisticated method based on a geometric analysis of the total charge density CHELPG another analysis method tries to reproduce calculated electrostatic potential with charges placed at the atom centers 10 Molecular orbital diagrams Direct analysis of the shapes and energies of molecular orbitals can often be useful for understanding molecular shapes and bonding Molecular properties often dictated by frontier molecular orbitals including HOMO highest occupied molecular orbital and LUMO lowest unoccupied molecular orbital Good resource is Albright Burdett and Wangbo Orbital Interactions in Chemistry Prof W F Schneider CBE 60547 7 Computational Chemistry 1010 9152009 122323 PM University of Notre Dame Fall 200 Lecture 4 Practical Aspects of Electronic Structure 4 u v quot 39 39 39 quot quot 39 39 Orbital analysis o en applied Wuhan semiempirical and DFT frameworks w Inn 1N0 c cc 4 u u Flam A Molecular orbual dilgmm showing 1 mam maul imme nnm um MuCl and 5mm fragmmls in both he slaggcrad left 6 and cclipsad rigm 6b cunrormaziuns Thc HOMO of u 1 1h 51 orbiml Inorg Chem 1989 18 3292 3296 Electronic Structure af Asymmetric Metal Melal Mulliple Bonds The dI dquot Complexes XAMo MMPHQ x 0HC1 Bruce E Bursmn and William F Schneidcr thmgd Navcmby 247 was m chum snnclurc nl camwun 39cunummx 11w Mo dlmanc unh willy forma ly usymmem Momqule chl39lc dmribuuon Frwuuga ed m the SC39ixrsw muhnd Such synems are knew at mixed phmphinrllkmmuipnd xy emj nar 1g 39 hibu 11 smnu ly mama sung r m1 1 mums mm 11 m m 11 1e 3 Izdllzandwniurmmmn hzsemmpkxmmmxmaronmlMrMatnplcbrmd Wuknmduwrh quota e 1 cm may oooc m mm ma m a m eclipsed mm conformnuon wim m Mmao Mild 9qu m mad m m dzmmmc mam aimcr mnvuuional murmur d1menpamlulnryhaxn m hxdemare phoxphlm mm m m a mum igand mnfnrrmuun m we drmdal mm 13fo w 1 5mm caccum ammumucmmy 1111 ansInna 12 23 23 m Ummmmmnm sznm Lecture 4 Practical Aspects of Electronic Structure What goes into an Electronic Structure Calculation 1 Choose structure BornOppenheimer approximation 2 Choose electronic structure method HF HFS 3 Choose electronic state a Charge electrons b Spinmultiplicity 4 Choose basis set and possible core potentials 5 Choose initial guess of orbital coefficients Cj a Determines initial charge density b Determines initial wavefunctions c Determines initial hamiltonian 6 Choose convergence criteria a Change in coefficients lt tolerance b Change in energy lt tolerance 7 Choose iteration scheme a Direct coefficient updatenot b Damped update c Optimizationbased i Firstorder steepest descents conjugate gradient ii Secondorder iii Extrapolation 1 DIISdirect inversion in the iterative subspace 8 Run 9 CHECKRESULTS Prof W F Schneider 9152009122323 PM CBE 60547 7 Computational Chemistry University of Notre Dame Fall 200 1212
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